Theorem mulnqprlemru 7133

Description: Lemma for mulnqpr 7136. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)

Assertion

Ref	Expression
mulnqprlemru	|- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Distinct variable groups:   A, l, u   B, l, u

Proof of Theorem mulnqprlemru

Dummy variables f g h r s t x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 7106	. . . . . 6 |- ( A e. Q. -> <. { l | l <Q A } , { u | A <Q u } >. e. P. )
2		nqprlu 7106	. . . . . 6 |- ( B e. Q. -> <. { l | l <Q B } , { u | B <Q u } >. e. P. )
3		df-imp 7028	. . . . . . 7 |- .P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g .Q h ) ) } >. )
4		mulclnq 6935	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
5	3, 4	genpelvu 7072	. . . . . 6 |- ( ( <. { l | l <Q A } , { u | A <Q u } >. e. P. /\ <. { l | l <Q B } , { u | B <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
6	1, 2, 5	syl2an 283	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) <-> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) ) )
7	6	biimpa 290	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) )
8		vex 2622	. . . . . . . . . . . . 13 |- s e. _V
9		breq2 3849	. . . . . . . . . . . . 13 |- ( u = s -> ( A <Q u <-> A <Q s ) )
10		ltnqex 7108	. . . . . . . . . . . . . 14 |- { l | l <Q A } e. _V
11		gtnqex 7109	. . . . . . . . . . . . . 14 |- { u | A <Q u } e. _V
12	10, 11	op2nd 5918	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) = { u | A <Q u }
13	8, 9, 12	elab2 2763	. . . . . . . . . . . 12 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) <-> A <Q s )
14	13	biimpi 118	. . . . . . . . . . 11 |- ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) -> A <Q s )
15	14	ad2antrl 474	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> A <Q s )
16	15	adantr 270	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> A <Q s )
17		vex 2622	. . . . . . . . . . . . 13 |- t e. _V
18		breq2 3849	. . . . . . . . . . . . 13 |- ( u = t -> ( B <Q u <-> B <Q t ) )
19		ltnqex 7108	. . . . . . . . . . . . . 14 |- { l | l <Q B } e. _V
20		gtnqex 7109	. . . . . . . . . . . . . 14 |- { u | B <Q u } e. _V
21	19, 20	op2nd 5918	. . . . . . . . . . . . 13 |- ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) = { u | B <Q u }
22	17, 18, 21	elab2 2763	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) <-> B <Q t )
23	22	biimpi 118	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) -> B <Q t )
24	23	ad2antll 475	. . . . . . . . . 10 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> B <Q t )
25	24	adantr 270	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> B <Q t )
26		ltrelnq 6924	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
27	26	brel 4490	. . . . . . . . . . 11 |- ( A <Q s -> ( A e. Q. /\ s e. Q. ) )
28	16, 27	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A e. Q. /\ s e. Q. ) )
29	26	brel 4490	. . . . . . . . . . 11 |- ( B <Q t -> ( B e. Q. /\ t e. Q. ) )
30	25, 29	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( B e. Q. /\ t e. Q. ) )
31		lt2mulnq 6964	. . . . . . . . . 10 |- ( ( ( A e. Q. /\ s e. Q. ) /\ ( B e. Q. /\ t e. Q. ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
32	28, 30, 31	syl2anc 403	. . . . . . . . 9 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A <Q s /\ B <Q t ) -> ( A .Q B ) <Q ( s .Q t ) ) )
33	16, 25, 32	mp2and 424	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q ( s .Q t ) )
34		breq2 3849	. . . . . . . . 9 |- ( r = ( s .Q t ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
35	34	adantl 271	. . . . . . . 8 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( ( A .Q B ) <Q r <-> ( A .Q B ) <Q ( s .Q t ) ) )
36	33, 35	mpbird 165	. . . . . . 7 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> ( A .Q B ) <Q r )
37		vex 2622	. . . . . . . 8 |- r e. _V
38		breq2 3849	. . . . . . . 8 |- ( u = r -> ( ( A .Q B ) <Q u <-> ( A .Q B ) <Q r ) )
39		ltnqex 7108	. . . . . . . . 9 |- { l | l <Q ( A .Q B ) } e. _V
40		gtnqex 7109	. . . . . . . . 9 |- { u | ( A .Q B ) <Q u } e. _V
41	39, 40	op2nd 5918	. . . . . . . 8 |- ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) = { u | ( A .Q B ) <Q u }
42	37, 38, 41	elab2 2763	. . . . . . 7 |- ( r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) <-> ( A .Q B ) <Q r )
43	36, 42	sylibr 132	. . . . . 6 |- ( ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ r = ( s .Q t ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
44	43	ex 113	. . . . 5 |- ( ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) /\ ( s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) /\ t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
45	44	rexlimdvva 2496	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` <. { l | l <Q A } , { u | A <Q u } >. ) E. t e. ( 2nd ` <. { l | l <Q B } , { u | B <Q u } >. ) r = ( s .Q t ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
46	7, 45	mpd 13	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )
47	46	ex 113	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( r e. ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) -> r e. ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) ) )
48	47	ssrdv 3031	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( 2nd ` ( <. { l | l <Q A } , { u | A <Q u } >. .P. <. { l | l <Q B } , { u | B <Q u } >. ) ) C_ ( 2nd ` <. { l | l <Q ( A .Q B ) } , { u | ( A .Q B ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074   E.wrex 2360   C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   2ndc2nd 5910   Q.cnq 6839   .Q cmq 6842   <Q cltq 6844   P.cnp 6850   .P. cmp 6853

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-inp 7025  df-imp 7028

This theorem is referenced by:  mulnqprlemfl  7134  mulnqpr  7136